Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
299 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,13,12,14,15,16)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(2,5)(6,8)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,3,4,7)(2,6,5,8)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,f^2=h,b^a=bd,c^a=ce,e^a=eg,f^c=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93301, #93558, #93683, … |
| maximal normal subgroup of | #93301, #93558, #93683, … |
| maximal quotient of | #99099, #99100, #99128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1687);